3D Shapes, Surface Area and Volume

What you will learn

• How to recognise cubes, cuboids and prisms.
• How to find volume using length × width × height or cross-section area × length.
• How to find surface area by adding the areas of all outside faces.
• How nets and face lists help you avoid missing a face.
• How to choose square units for surface area and cubic units for volume.
• How to work backwards from a volume or combine shapes in a practical context.

Why this matters

Surface area and volume appear in packaging, containers, storage, paint, wrapping, concrete, density and capacity problems. A good face-by-face method stops 3D questions feeling like guesswork.

Prior knowledge

You should already be comfortable with:

• finding the area of rectangles and triangles,
• multiplying three numbers,
• using formulae carefully,
• knowing that cm² is an area unit and cm³ is a volume unit.

Clear explanation

Volume means space inside

Volume measures how much 3D space a solid takes up or can hold. For a cuboid, multiply the three perpendicular lengths.

For any prism, find the area of the constant cross-section, then multiply by the length of the prism.

Surface area means outside faces

Surface area is the total area of every outside face. For a cuboid, there are three pairs of matching faces: front and back, top and bottom, left and right.

You do not have to memorise that formula if a face list is clearer. Add every outside face once.

Capacity questions use the same volume idea but may change units. Remember that 1 litre = 1000 cm³, so convert before comparing containers, liquids or costs.

In reverse questions, write the normal formula first and then divide by the known parts. In compound-solid questions, split the solid into simple cuboids or prisms, find each volume or face area, and then combine only the parts the question asks for.

Worked examples

Quick checks

Choose an answer, then check your thinking.

Practice questions

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For 3D shape questions, marks usually come from choosing whether the task is surface area, volume or capacity, listing the right outside faces or cross-section, substituting dimensions accurately, using inverse operations for missing lengths, and writing square units for surface area or cubic units for volume.

Common mistakes

• Using area units for volume: volume needs cubic units such as cm³ or m³.
• Missing hidden matching faces: a closed cuboid has six faces, not just the three you can see.
• Multiplying all numbers for surface area: surface area is found by adding face areas.
• Using the wrong triangle height: in a triangular prism, the triangle height must be perpendicular to the base.

Extension challenge

A cuboid has dimensions 12 cm, 5 cm and h cm. Its volume is 420 cm³. Find h and then find the surface area.

Exam-board guidance

Surface area and volume are core GCSE Maths skills across all boards. The common exam habit is to decide whether the question asks for outside area or inside space before starting any calculations.

Next lesson

Next, continue Geometry and Measures with Units, Metric Conversions and Measures.